Between a beam and catenary: Influence of geometry on gravitational stresses and stability of natural rock arches

Abstract

The self-weight of freestanding natural arches imparts stresses on the lintel and abutments, setting the conditions for rock fracture and erosion. Rock is weak in tension, therefore an ideal arch form is one that minimizes tensile stresses and supports the weight of the lintel in compression, e.g. an inverted catenary. However, rock mass structure, including bedding, cross-bedding and other discontinuities, often imparts strong control on the geometry of arches, leading to forms that are less favorable from a stress perspective. Here we analyze a suite of nineteen arch models created from ground- and drone-based photogrammetry to assess the three-dimensional static stress field under gravitational loading. The models represent a range of arch lengths from 4 to 88 m, as well as a variety of forms, and use material properties previously calibrated from dynamic analysis of ambient vibrations. Our results demonstrate that arches shaped like beams have relatively high tensile stresses and a nearly symmetrical statistical distribution of principal stresses, while those with convex forms have comparably lower tensile stresses and statistical distributions favoring compressive stresses. In-situ observations of tensile cracks frequently correspond to the location of predicted tensile stresses in our models. We calculated the ratio of mean principal stresses for each arch, which is theoretically 1 for a flat prismatic beam and approaches infinity for an ideal inverted catenary. We found several arches with mean principal stress ratio around 1 and that values increased with lintel convexity. These results indicate that while self-sculpture might attempt to create ideal stress-based forms, discontinuities can control lintel geometry and natural arches evolve with forms that may be less favorable for long-term stability. The mean principal stress ratio is a simple metric to classify and compare arches, useful for assessment of arch stability supporting conservation and hazard analyses.